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A102437 Let pi be an unrestricted partition of n with the summands written in binary notation. a(n) is the number of such partitions whose binary representation has an odd number of binary ones.

%I A102437 #25 Jul 25 2018 10:58:13
%S A102437 0,1,1,1,3,3,5,9,10,14,22,28,37,53,66,85,120,147,188,252,308,394,509,
%T A102437 621,783,990,1210,1500,1872,2272,2793,3447,4152,5064,6184,7414,8984,
%U A102437 10856,12964,15592,18711,22250,26576,31690,37520,44565,52856,62292
%N A102437 Let pi be an unrestricted partition of n with the summands written in binary notation. a(n) is the number of such partitions whose binary representation has an odd number of binary ones.
%H A102437 Alois P. Heinz, <a href="/A102437/b102437.txt">Table of n, a(n) for n = 0..1000</a>
%e A102437 a(5) = 3 because there are 3 partitions of 5 with an odd number of binary ones in their binary representation, namely: 11+10, 10+10+1 and 1+1+1+1+1.
%p A102437 p:= proc(n) option remember; local c, m;
%p A102437       c:= 0; m:= n;
%p A102437       while m>0 do c:= c +irem(m, 2, 'm') od;
%p A102437       c
%p A102437     end:
%p A102437 b:= proc(n, i, t) option remember;
%p A102437       if n<0 then 0
%p A102437     elif n=0 then t
%p A102437     elif i=0 then 0
%p A102437     else b(n, i-1, t) +b(n-i, i, irem(p(i)+t, 2))
%p A102437       fi
%p A102437     end:
%p A102437 a:= n-> b(n, n, 0):
%p A102437 seq(a(n), n=0..60);  # _Alois P. Heinz_, Feb 21 2011
%t A102437 Table[Length[Select[Map[Apply[Join,#]&,Map[IntegerDigits[#,2]&,Partitions[n]]],OddQ[Count[#,1]]&]],{n,0,40}] (* _Geoffrey Critzer_, Sep 28 2013 *)
%o A102437 (PARI) seq(n)={apply(t->polcoeff(lift(t), 1), Vec(prod(i=1, n, 1/(1 - x^i*Mod( y^hammingweight(i), y^2-1 )) + O(x*x^n))))} \\ _Andrew Howroyd_, Jul 20 2018
%Y A102437 Cf. A000041, A000120, A102425, A316996.
%K A102437 nonn
%O A102437 0,5
%A A102437 _David S. Newman_, Feb 23 2005
%E A102437 More terms from _Vladeta Jovovic_, Feb 23 2005